We consider a random walk $S$ in the domain of attraction of a standard normal law $Z$, \textit{ie} there exists a positive sequence $a_n$ such that $S_n/a_n$ converges in law towards $Z$. The main result of this note is that the rescaled process $(S_{\lfloor nt \rfloor}/a_n, t \geq 0)$ conditioned to stay non-negative, to start and to come back \textit{near the origin} converges in law towards the normalized brownian excursion.
In this paper we apply the spectral theory of linear diffusions to study the one-dimensional Liouville Brownian Motion and Liouville Brownian excursions from a given point. As an application we estimate the fractal dimensions of level sets of one-dimensional Liouville Brownian motion as well as various probabilistic asymptotic behaviours of Liouville Brownian motion and Liouville Brownian excursions.
We study properties of occupation times by Brownian excursions and Brownian loops in two-dimensional domains. This allows for instance to interpret some Gaussian fields, such as the Gaussian Free Fields as (properly normalized) fluctuations of the total occupation time of a Poisson cloud of Brownian excursions when the intensity of the cloud goes to infinity.
We study the density of the time average of the Brownian meander/excursion over the time interval [0,1]. Moreover we give an expression for the Brownian meander/excursion conditioned to have a fixed time average.
In this paper, following earlier results in [2] we derive the asymptotic distribution as $t \to \infty$, of the excursion of Brownian motion straddling $t$, into an interval $(a,b)$, conditional on the event that there is such an excursion.
We develop an excursion theory for Brownian motion indexed by the Brownian tree, which in many respects is analogous to the classical Itô theory for linear Brownian motion. Each excursion is associated with a connected component of the complement of the zero set of the tree-indexed Brownian motion. Each such connectedcomponent is itself a continuous tree, and we introduce a quantity measuring the length of its boundary. The collection of boundary lengths coincides with the collection of jumps of a continuous-state branching process with branching mechanism $ψ(u)=\sqrt{8/3}\,u^{3/2}$. Furthermo…
By using the law of the excursions of Brownian motion with drift, we find the distribution of the $n-$th passage time of Brownian motion through a straight line $S(t)= a + bt.$ In the special case when $b = 0,$ we extend the result to a space-time transformation of Brownian motion.
The aim of this paper is to present a self-similar growth-fragmentation process linked to a Brownian excursion in the upper half-plane $\mathbb{H}$, obtained by cutting the excursion at horizontal levels. We prove that the associated growth-fragmentation is related to one of the growth-fragmentation processes introduced by Bertoin, Budd, Curien and Kortchemski.
Functionals of Brownian bridge arise as limiting distributions in nonparametric statistics. In this paper we will give a derivation of distributions of extrema of the Brownian bridge based on excursion theory for Brownian motion. Only the Poisson character of the excursion process will be used. Particular cases of calculations include the distributions of the Kolmogorov-Smirnov statistic, the Kuiper statistic, and the ratio of the maximum positive ordinate to the minumum negative ordinate.
We consider the Brownian interlacements model in Euclidean space, introduced by A.S. Sznitman in \cite{sznitman2013scaling}. We give estimates for the asymptotics of the visibility in the vacant set. We also consider visibility inside the vacant set of the Brownian excursion process in the unit disc and show that it undergoes a phase transition regarding visibility to infinity as in \cite{benjamini2009visibility}. Additionally, we determine the critical value and that there is no visibility to infinity at the critical intensity.
Several Brownian areas are considered in this paper: the Brownian excursion area, the Brownian bridge area, the Brownian motion area, the Brownian meander area, the Brownian double meander area, the positive part of Brownian bridge area, the positive part of Brownian motion area. We are interested in the asymptotics of the right tail of their density function. Inverting a double Laplace transform, we can derive, in a mechanical way, all terms of an asymptotic expansion. We illustrate our technique with the computation of the first four terms. We also obtain asymptotics for the right tail of th…
The integrated Brownian motion is sometimes known as the Langevin process. Lachal studied several excursion laws induced by the latter. Here we follow a different point of view developed by Pitman for general stationary processes. We first construct a stationary Langevin process and then determine explicitly its stationary excursion measure. This is then used to provide new descriptions of Ito's excursion measure of the Langevin process reflected at a completely inelastic boundary, which has been introduced recently by Bertoin.
We show the result that is stated in the title of the paper, which has consequences about decomposition of Brownian loop-soup clusters in two dimensions.
The Laplace transform of the $d$-dimensional distribution of Brownian excursion is expressed as the Laplace transform of the $(d+1)$-dimensional distribution of an auxiliary Markov process, started from a $σ$-finite measure and with the roles of arguments and times interchanged. A similar identity holds for the Laplace transform of a generalized meander, which is expressed as the Laplace transform of the same auxiliary Markov process, with a different initial law.
We consider the problem of conditioning the Brownian excursion to have a fixed time average over the interval [0,1] and we study an associated stochastic partial differential equation with reflection at 0 and with the constraint of conservation of the space average. The equation is driven by the derivative in space of a space-time white noise and contains a double Laplacian in the drift. Due to the lack of the maximum principle for the double Laplacian, the standard techniques based on the penalization method do not yield existence of a solution.
We consider two jointly stationary and ergodic random measures $ξ$ and $η$ on the real line $\mathbb{R}$ with equal intensities. An allocation is an equivariant random mapping from $\mathbb{R}$ to $\mathbb{R}$. We give sufficient and partially necessary conditions for the existence of allocations transporting $ξ$ to $η$. An important ingredient of our approach is to introduce a transport kernel balancing $ξ$ and $η$, provided these random measures are mutually singular. In the second part of the paper, we apply this result to the path decomposition of a two-sided Brownian motion into three ind…
The Brownian excursion measure is a conformally invariant infinite measure on curves. It figured prominently in one of the first major applications of SLE, namely the explicit calculations of the planar Brownian intersection exponents from which the Hausdorff dimension of the frontier of the Brownian path could be computed (Lawler, Schramm, and Werner, 2001). In this paper we define the simple random walk excursion measure and show that for any bounded, simply connected Jordan domain D, the simple random walk excursion measure on D converges in the scaling limit to the Brownian excursion measu…
Let $\{B(t), t \geq 0\}$ be a standard Brownian motion in $\mathbb{R}$. Let $T$ be the first return time to 0 after hitting 1, and $\{L(T,x), x \in \mathbb{R}\}$ be the local time process at time $T$ and level $x$. The distribution of $L(T,x)$ for each $x \in \mathbb{R}$ is determined. This is applied to the estimation of a $L^1$ integral on $\mathbb{R}$.
We consider the process of $n$ Brownian excursions conditioned to be nonintersecting. We show the distribution functions for the top curve and the bottom curve are equal to Fredholm determinants whose kernel we give explicitly. In the simplest case, these determinants are expressible in terms of Painlevé V functions. We prove that as $n\to \infty$, the distributional limit of the bottom curve is the Bessel process with parameter 1/2. (This is the Bessel process associated with Dyson's Brownian motion.) We apply these results to study the expected area under the bottom and top curves.
This survey is a collection of various results and formulas by different authors on the areas (integrals) of five related processes, viz.\spacefactor =1000 Brownian motion, bridge, excursion, meander and double meander; for the Brownian motion and bridge, which take both positive and negative values, we consider both the integral of the absolute value and the integral of the positive (or negative) part. This gives us seven related positive random variables, for which we study, in particular, formulas for moments and Laplace transforms; we also give (in many cases) series representations and as…
This note re-addresses the Paris barrier options proposed by Yor and collaborators and their valuation using the Laplace transform approach. The notion of Paris barrier options, based on excursion theory and using the Brownian meander, is extended such that their valuation is now possible at any point during their lifespan. The pertinent Laplace transforms are modified when necessary.
Permutations that avoid given patterns are among the most classical objects in combinatorics and have strong connections to many fields of mathematics, computer science and biology. In this paper we study the scaling limits of a random permutation avoiding a pattern of length 3 and their relations to Brownian excursion. Exploring this connection to Brownian excursion allows us to strengthen the recent results of Madras and Pehlivan, and Miner and Pak as well as to understand many of the interesting phenomena that had previously gone unexplained.
A notion of convergence of excursion measures is introduced. It is proved that convergence of excursion measures implies convergence in law of the processes pieced together from excursions. This result is applied to obtain homogenization theorems of jumping-in extensions for positive self-similar Markov processes, for Walsh diffusions and for the Brownian motion on the Sierpiński gasket.
This paper reviews known results which connect Riemann's integral representations of his zeta function, involving Jacobi's theta function and its derivatives, to some particular probability laws governing sums of independent exponential variables. These laws are related to one-dimensional Brownian motion and to higher dimensional Bessel processes. We present some characterizations of these probability laws, and some approximations of Riemann's zeta function which are related to these laws.
We consider a super-Brownian motion $X$. Its canonical measures can be studied through the path-valued process called the Brownian snake. We obtain the limiting behavior of the volume of the $ε$-neighborhood for the range of the Brownian snake, and as a consequence we derive the analogous result for the range of super-Brownian motion and for the support of the integrated super-Brownian excursion. Then we prove the support of $X_t$ is capacity-equivalent to $[0,1]^2$ in $\R^d$, $d\geq 3$, and the range of $X$, as well as the support of the integrated super-Brownian excursion are capacity-equiva…
Permutations that avoid given patterns are among the most classical objects in combinatorics and have strong connections to many fields of mathematics, computer science and biology. In this paper we study fixed points of both 123- and 231-avoiding permutations. We find an exact description for a scaling limit of the empirical distribution of fixed points in term of Brownian excursion. This builds on the connections between pattern-avoiding permutations and Brownian excursion developed in Part I of this series and strengthens the recent results of Elizalde (2012) and Miner and Pak (2014) on fix…
Area fluctuations of a Brownian excursion are described by the Airy distribution, which found applications in different areas of physics, mathematics and computer science. Here we generalize this distribution to describe the area fluctuations on a \emph{subinterval} of a Brownian excursion. In the first version of the problem (Model 1) no additional conditions are imposed. In the second version (Model 2) we study the distribution of the area fluctuations on a subinterval given the excursion area on the whole interval. Both versions admit convenient path-integral formulations. In Model 1 we obt…
Excursion reflected Brownian motion (ERBM) is a strong Markov process defined in a finitely connected domain $D \subset \mathbb{C}$ that behaves like a Brownian motion away from the boundary of $D$ and picks a point according to harmonic measure from infinity to reflect from every time it hits a boundary component. We give a construction of ERBM using its conformal invariance and develop the basic theory of its harmonic functions. One important reason for studying ERBM is the hope that it will be a useful tool in the study of SLE in multiply connected domains. To this end, we develop the basic…
In a previous paper, we established strong existence and uniqueness for a reflected diffusion $(X,S)$ with values in $\bar D\times \mathbbm{R}^p$, solving the following pair of stochastic differential equations: $$ dX_t = σ(X_t)dB_t + \vecγ(X_t,S_t)dL_t, $$ $$ dS_t = [\vec{g}(X_t) - S_t ] dL_t. $$ Here $L_t$ is the boundary local time of $X_t$, and $\vecγ$ points uniformly into the domain $D$. The process $(X,S)$ is called spinning Brownian motion (sBm). In this article, we prove uniqueness of the stationary distribution of spinning Brownian motion by studying excursions away from the boundary…
We study limit distributions for random variables defined in terms of coefficients of a power series which is determined by a certain linear functional equation. Our technique combines the method of moments with the kernel method of algebraic combinatorics. As limiting distributions the area distributions of the Brownian excursion and meander occur. As combinatorial applications we compute the area laws for discrete excursions and meanders with an arbitrary finite set of steps and the area distribution of column convex polyominoes. As a by-product of our approach we find the joint distribution…
Excursion reflected Brownian motion (ERBM) is a strong Markov process defined in a finitely connected domain $D \subset \C$ that behaves like a Brownian motion away from the boundary of $D$ and picks a point according to harmonic measure from infinity to reflect from every time it hits a boundary component. We give a new construction of ERBM using its conformal invariance and discuss the relationship between the Poisson kernel and Green's function for ERBM and conformal maps into certain classes of finitely connected domains. One important reason for studying ERBM is the hope that it will be a…
Trees in Brownian excursions have been studied since the late 1980s. Forests in excursions of Brownian motion above its past minimum are a natural extension of this notion. In this paper we study a forest-valued Markov process which describes the growth of the Brownian forest. The key result is a composition rule for binary Galton--Watson forests with i.i.d. exponential branch lengths. We give elementary proofs of this composition rule and explain how it is intimately linked with Williams' decomposition for Brownian motion with drift.
In this paper, we consider hashing with linear probing for a hashing table with m places, n items (n < m), and l = m<n empty places. For a non computer science-minded reader, we shall use the metaphore of n cars parking on m places: each car chooses a place at random, and if this place k is occupied, the car tries successively k+1, k+2, ... until it finds an empty place (with the convention that place m+1 is actually place 1). Pittel [42] proves that when l/m goes to some positive limit a < 1, the size of the largest block of consecutive cars is O(log m). In this paper we examine at which leve…
It is proved that generalized excursion measures can be constructed via time change of Ito's Brownian excursion measure. A tightness-like condition on strings is introduced to prove a convergence theorem of generalized excursion measures. The convergence theorem is applied to obtain a conditional limit theorem, a kind of invariance principle where the limit is the Bessel meander.
We consider regenerative processes with values in some Polish space. We define their ε-big excursions as excursions e such that f(e)>ε, where f is some given functional on the space of excursions which can be thought of as, e.g., the length or the height of e. We establish a general condition that guarantees the convergence of a sequence of regenerative processes involving the convergence of ε-big excursions and of their endpoints, for all εin a countable set whose closure contains 0. Finally, we provide various sufficient conditions on the excursion measures of this sequence for this general …
Invariance principles are obtained for a Markov process on a half-line with continuous paths on the interior. The domains of attraction of the two different types of self-similar processes are investigated. Our approach is to establish convergence of excursion point processes, which is based on Itô's excursion theory and a recent result on convergence of excursion measures by Fitzsimmons and the present author.
We derive several explicit distributions of functionals of Brownian motion indexed by the Brownian tree. In particular, we give a direct proof of a result of Bousquet-Mélou and Janson identifying the distribution of the density at 0 of the integrated super-Brownian excursion.
It is known that after scaling a random Motzkin path converges to a Brownian excursion. We prove that the fluctuations of the counting processes of the ascent steps, the descent steps and the level steps converge jointly to linear combinations of two independent processes: a Brownian motion and a Brownian excursion. The proofs rely on the Laplace transforms and an integral representation based on an identity connecting non-crossing pair partitions and joint moments of an explicit non-homogeneous Markov process.
We study the convergence of the $M/G/1$ processor-sharing, queue length process in the heavy traffic regime, in the finite variance case. To do so, we combine results pertaining to Lévy processes, branching processes and queuing theory. These results yield the convergence of long excursions of the queue length processes, toward excursions obtained from those of some reflected Brownian motion with drift, after taking the image of their local time process by the Lamperti transformation. We also show, via excursion theoretic arguments, that this entails the convergence of the entire processes to …
We provide a new construction of the Brownian disks, which have been defined by Bettinelli and Miermont as scaling limits of quadrangulations with a boundary when the boundary size tends to infinity. Our method is very similar to the construction of the Brownian map, but it makes use of the positive excursion measure of the Brownian snake which has been introduced recently. This excursion measure involves a random continuous tree whose vertices are assigned nonnegative labels, which correspond to distances from the boundary in our approach to the Brownian disk. We provide several applications …
We prove that a sequence of semi-discrete approximations converges to a multiplicative functional for reflected Brownian motion, which intuitively represents the Lyapunov exponent for the corresponding stochastic flow. The method of proof is based on a study of the deterministic version of the problem and the excursion theory.
We consider Kallenberg's hypothesis on the characteristic function of a Lévy process and show that it allows the construction of weakly continuous bridges of the Lévy process conditioned to stay positive. We therefore provide a notion of normalized excursions Lévy processes above their cumulative minimum. Our main contribution is the construction of a continuous version of the transition density of the Lévy process conditioned to stay positive by using the weakly continuous bridges of the Lévy process itself. For this, we rely on a method due to Hunt which had only been shown to provide upper …
We show that the range of a long Brownian bridge in the hyperbolic space converges after suitable renormalisation to the Brownian continuum random tree. This result is a relatively elementary consequence of $\bullet$ A theorem by Bougerol and Jeulin, stating that the rescaled radial process converges to the normalized Brownian excursion, $\bullet$ A property of invariance under re-rooting, $\bullet$ The hyperbolicity of the ambient space in the sense of Gromov. A similar result is obtained for the rescaled infinite Brownian loop in hyperbolic space.
We consider systems of interacting diffusions with local population regulation. Our main result shows that the total mass process of such a system is bounded above by the total mass process of a tree of excursions with appropriate drift and diffusion coefficients. As a corollary, this entails a sufficient, explicit condition for extinction of the total mass as time tends to infinity. On the way to our comparison result, we establish that systems of interacting diffusions with uniform migration between finitely many islands converge to a tree of excursions as the number of islands tends to infi…
This article presents a complete second order theory for a large class of geometric functionals on homogeneous Poisson input. In particular, the results don't require the existence of a radius of stabilisation. Hence they can be applied to geometric functionals of spatial shot-noise fields excursions such as volume, perimeter, or Euler characteristic (the method still applies to stabilising functionals). More generally, it must be checked that a local contribution to the functional is not strongly affected under a perturbation of the input far away. In this case the exact asymptotic variance i…
In a recent paper by Gorin and Shkolnikov (2016), they have found, as a corollary to their result relevant to random matrix theory, that the area below a normalized Brownian excursion minus one half of the integral of the square of its total local time, is identical in law with a centered Gaussian random variable with variance $1/12$. In this note, we give a pathwise interpretation to their identity; Jeulin's identity connecting normalized Brownian excursion and its local time plays an essential role in the exposition.
We find exact small deviation asymptotics with respect to weighted Hilbert norm for some well-known Gaussian processes. Our approach does not require the knowledge of eigenfunctions of the covariance operator of a weighted process. Such a peculiarity of the method makes it possible to generalize many previous results in this area. We also obtain new relations connected to exact small deviation asymptotics for a Brownian excursion, a Brownian meander, and Bessel processes and bridges.
Let $(\mathcal{T},d)$ be the random real tree with root $ρ$ coded by a Brownian excursion. So $(\mathcal{T},d)$ is (up to normalisation) Aldous CRT \cite{AldousI} (see Le Gall \cite{LG91}). The $a$-level set of $\mathcal{T}$ is the set $\mathcal{T}(a)$ of all points in $\mathcal{T}$ that are at distance $a$ from the root. We know from Duquesne and Le Gall \cite{DuLG06} that for any fixed $a\in (0, \infty)$, the measure $\ell^a$ that is induced on $\mathcal{T}(a)$ by the local time at $a$ of the Brownian excursion, is equal, up to a multiplicative constant, to the Hausdorff measure in $\mathcal…
A superprocess with coalescing spatial motion is constructed in terms of one-dimensional excursions. Based on this construction, it is proved that the superprocess is purely atomic and arises as scaling limit of a special form of the superprocess with dependent spatial motion studied in Dawson {\it et al.} (2001) and Wang (1997, 1998).
Consider a branching random walk on the real line. Madaule showed the renormalized trajectory of an individual selected according to the critical Gibbs measure converges in law to a Brownian meander. Besides, Chen proved that the renormalized trajectory leading to the leftmost individual at time $n$ converges in law to a standard Brownian excursion. In this article, we prove that the renormalized trajectory of an individual selected according to a supercritical Gibbs measure also converges in law toward the Brownian excursion. Moreover, refinements of this results enables to express the probab…
By computations on generating functions, Szekeres proved in 1983 that the law of the diameter of a uniformly distributed rooted labelled tree with n vertices, rescaled by a factor n^{1/2} , converges to a distribution whose density is explicit. Aldous observed in 1991 that this limiting distribution is the law of the diameter of the Brownian tree. In our article, we provide a computation of this law which is directly based on the normalized Brownian excursion. Moreover, we provide an explicit formula for the joint law of the height and diameter of the Brownian tree, which is a new result.
The Brownian separable permuton is a random probability measure on the unit square, which was introduced by Bassino, Bouvel, Féray, Gerin, Pierrot (2016) as the scaling limit of the diagram of the uniform separable permutation as size grows to infinity. We show that, almost surely, the permuton is the pushforward of the Lebesgue measure on the graph of a random measure-preserving function associated to a Brownian excursion whose strict local minima are decorated with i.i.d. signs. As a consequence, its support is almost surely totally disconnected, has Hausdorff dimension one, and enjoys self-…
We give a short proof that a uniform noncrossing partition of the regular $n$-gon weakly converges toward Aldous's Brownian triangulation of the disk, in the sense of the Hausdorff topology. This result was first obtained by Curien & Kortchemski, using a more complicated encoding. Thanks to a result of Marchal on strong convergence of Dyck paths toward the Brownian excursion, we furthermore give an algorithm that allows to recursively construct a sequence of uniform noncrossing partitions for which the previous convergence holds almost surely. In addition, we also treat the case of uniform non…
